# Focal length

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The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative focal length indicates that the system diverges light. A system with a shorter focal length bends the rays more sharply, bringing them to a focus in a shorter distance or diverging them more quickly. For the special case of a thin lens in air, a positive focal length is the distance over which initially collimated (parallel) rays are brought to a focus, or alternatively a negative focal length indicates how far in front of the lens a point source must be located to form a collimated beam. For more general optical systems, the focal length has no intuitive meaning; it is simply the inverse of the system's optical power.

## Contents

In most photography and all telescopy, where the subject is essentially infinitely far away, longer focal length (lower optical power) leads to higher magnification and a narrower angle of view; conversely, shorter focal length or higher optical power is associated with lower magnification and a wider angle of view. On the other hand, in applications such as microscopy in which magnification is achieved by bringing the object close to the lens, a shorter focal length (higher optical power) leads to higher magnification because the subject can be brought closer to the center of projection.

## Thin lens approximation

For a thin lens in air, the focal length is the distance from the center of the lens to the principal foci (or focal points) of the lens. For a converging lens (for example a convex lens), the focal length is positive and is the distance at which a beam of collimated light will be focused to a single spot. For a diverging lens (for example a concave lens), the focal length is negative and is the distance to the point from which a collimated beam appears to be diverging after passing through the lens.

When a lens is used to form an image of some object, the distance from the object to the lens u, the distance from the lens to the image v, and the focal length f are related by

${\displaystyle {\frac {1}{f}}={\frac {1}{u}}+{\frac {1}{v}}\ .}$

The focal length of a thin convex lens can be easily measured by using it to form an image of a distant light source on a screen. The lens is moved until a sharp image is formed on the screen. In this case 1/u is negligible, and the focal length is then given by

${\displaystyle f\approx v\ .}$

Determining the focal length of a concave lens is somewhat more difficult. The focal length of such a lens is considered that point at which the spreading beams of light would meet before the lens if the lens were not there. No image is formed during such a test, and the focal length must be determined by passing light (for example, the light of a laser beam) through the lens, examining how much that light becomes dispersed/ bent, and following the beam of light backwards to the lens's focal point.

## General optical systems

For a thick lens (one which has a non-negligible thickness), or an imaging system consisting of several lenses or mirrors (e.g. a photographic lens or a telescope), the focal length is often called the effective focal length (EFL), to distinguish it from other commonly used parameters:

• Front focal length (FFL) or front focal distance (FFD) (sF) is the distance from the front focal point of the system (F) to the vertex of the first optical surface (S1). [1] [2]
• Back focal length (BFL) or back focal distance (BFD) (s′F′) is the distance from the vertex of the last optical surface of the system (S2) to the rear focal point (F′). [1] [2]

For an optical system in air, the effective focal length (f and f′) gives the distance from the front and rear principal planes (H and H′) to the corresponding focal points (F and F′). If the surrounding medium is not air, then the distance is multiplied by the refractive index of the medium (n is the refractive index of the substance from which the lens itself is made; n1 is the refractive index of any medium in front of the lens; n2 is that of any medium in back of it). Some authors call these distances the front/rear focal lengths, distinguishing them from the front/rear focal distances, defined above. [1]

In general, the focal length or EFL is the value that describes the ability of the optical system to focus light, and is the value used to calculate the magnification of the system. The other parameters are used in determining where an image will be formed for a given object position.

For the case of a lens of thickness d in air (n1 = n2 = 1), and surfaces with radii of curvature R1 and R2, the effective focal length f is given by the Lensmaker's equation:

${\displaystyle {\frac {1}{f}}=(n-1)\left({\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}+{\frac {(n-1)d}{nR_{1}R_{2}}}\right),}$

where n is the refractive index of the lens medium. The quantity 1/f is also known as the optical power of the lens.

The corresponding front focal distance is: [3]

${\displaystyle {\mbox{FFD}}=f\left(1+{\frac {(n-1)d}{nR_{2}}}\right),}$

and the back focal distance:

${\displaystyle {\mbox{BFD}}=f\left(1-{\frac {(n-1)d}{nR_{1}}}\right).}$

In the sign convention used here, the value of R1 will be positive if the first lens surface is convex, and negative if it is concave. The value of R2 is negative if the second surface is convex, and positive if concave. Note that sign conventions vary between different authors, which results in different forms of these equations depending on the convention used.

For a spherically curved mirror in air, the magnitude of the focal length is equal to the radius of curvature of the mirror divided by two. The focal length is positive for a concave mirror, and negative for a convex mirror. In the sign convention used in optical design, a concave mirror has negative radius of curvature, so

${\displaystyle f=-{R \over 2},}$

where R is the radius of curvature of the mirror's surface.

## In photography

28 mm lens
50 mm lens
70 mm lens
210 mm lens
An example of how lens choice affects angle of view. The photos above were taken by a 35 mm camera at a fixed distance from the subject.

Camera lens focal lengths are usually specified in millimetres (mm), but some older lenses are marked in centimetres (cm) or inches.

Focal length (f) and field of view (FOV) of a lens are inversely proportional. For a standard rectilinear lens, FOV = 2 arctan x/2f, where x is the diagonal of the film.

When a photographic lens is set to "infinity", its rear nodal point is separated from the sensor or film, at the focal plane, by the lens's focal length. Objects far away from the camera then produce sharp images on the sensor or film, which is also at the image plane.

To render closer objects in sharp focus, the lens must be adjusted to increase the distance between the rear nodal point and the film, to put the film at the image plane. The focal length (f), the distance from the front nodal point to the object to photograph (s1), and the distance from the rear nodal point to the image plane (s2) are then related by:

${\displaystyle {\frac {1}{s_{1}}}+{\frac {1}{s_{2}}}={\frac {1}{f}}.}$

As s1 is decreased, s2 must be increased. For example, consider a normal lens for a 35 mm camera with a focal length of f = 50 mm. To focus a distant object (s1  ∞), the rear nodal point of the lens must be located a distance s2 = 50 mm from the image plane. To focus an object 1 m away (s1 = 1,000 mm), the lens must be moved 2.6 mm farther away from the image plane, to s2 = 52.6 mm.

The focal length of a lens determines the magnification at which it images distant objects. It is equal to the distance between the image plane and a pinhole that images distant objects the same size as the lens in question. For rectilinear lenses (that is, with no image distortion), the imaging of distant objects is well modelled as a pinhole camera model. [4] This model leads to the simple geometric model that photographers use for computing the angle of view of a camera; in this case, the angle of view depends only on the ratio of focal length to film size. In general, the angle of view depends also on the distortion. [5]

A lens with a focal length about equal to the diagonal size of the film or sensor format is known as a normal lens; its angle of view is similar to the angle subtended by a large-enough print viewed at a typical viewing distance of the print diagonal, which therefore yields a normal perspective when viewing the print; [6] this angle of view is about 53 degrees diagonally. For full-frame 35 mm-format cameras, the diagonal is 43 mm and a typical "normal" lens has a 50 mm focal length. A lens with a focal length shorter than normal is often referred to as a wide-angle lens (typically 35 mm and less, for 35 mm-format cameras), while a lens significantly longer than normal may be referred to as a telephoto lens (typically 85 mm and more, for 35 mm-format cameras). Technically, long focal length lenses are only "telephoto" if the focal length is longer than the physical length of the lens, but the term is often used to describe any long focal length lens.

Due to the popularity of the 35 mm standard, camera–lens combinations are often described in terms of their 35 mm-equivalent focal length, that is, the focal length of a lens that would have the same angle of view, or field of view, if used on a full-frame 35 mm camera. Use of a 35 mm-equivalent focal length is particularly common with digital cameras, which often use sensors smaller than 35 mm film, and so require correspondingly shorter focal lengths to achieve a given angle of view, by a factor known as the crop factor.

## Related Research Articles

In optics, aberration is a property of optical systems such as lenses that causes light to be spread out over some region of space rather than focused to a point. Aberrations cause the image formed by a lens to be blurred or distorted, with the nature of the distortion depending on the type of aberration. Aberration can be defined as a departure of the performance of an optical system from the predictions of paraxial optics. In an imaging system, it occurs when light from one point of an object does not converge into a single point after transmission through the system. Aberrations occur because the simple paraxial theory is not a completely accurate model of the effect of an optical system on light, rather than due to flaws in the optical elements.

For many cameras, depth of field (DOF) is the distance between the nearest and the farthest objects that are in acceptably sharp focus in an image. The depth of field can be calculated based on focal length, distance to subject, the acceptable circle of confusion size, and aperture. A particular depth of field may be chosen for technical or artistic purposes. Limitations of depth of field can sometimes be overcome with various techniques/equipment.

A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (elements), usually arranged along a common axis. Lenses are made from materials such as glass or plastic, and are ground and polished or molded to a desired shape. A lens can focus light to form an image, unlike a prism, which refracts light without focusing. Devices that similarly focus or disperse waves and radiation other than visible light are also called lenses, such as microwave lenses, electron lenses, acoustic lenses, or explosive lenses.

In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the property that it is constant for a beam as it goes from one material to another, provided there is no refractive power at the interface. The exact definition of the term varies slightly between different areas of optics. Numerical aperture is commonly used in microscopy to describe the acceptance cone of an objective, and in fiber optics, in which it describes the range of angles within which light that is incident on the fiber will be transmitted along it.

In optics, the f-number of an optical system such as a camera lens is the ratio of the system's focal length to the diameter of the entrance pupil. It is also known as the focal ratio, f-ratio, or f-stop, and is very important in photography. It is a dimensionless number that is a quantitative measure of lens speed; increasing the f-number is referred to as stopping down. The f-number is commonly indicated using a lower-case hooked f with the format f/N, where N is the f-number.

In optics, a circle of confusion is an optical spot caused by a cone of light rays from a lens not coming to a perfect focus when imaging a point source. It is also known as disk of confusion, circle of indistinctness, blur circle, or blur spot.

The angle of view is the decisive variable for the visual perception of the size or projection of the size of an object.

A camera lens is an optical lens or assembly of lenses used in conjunction with a camera body and mechanism to make images of objects either on photographic film or on other media capable of storing an image chemically or electronically.

An optical telescope is a telescope that gathers and focuses light, mainly from the visible part of the electromagnetic spectrum, to create a magnified image for direct view, or to make a photograph, or to collect data through electronic image sensors.

In photography and cinematography, perspective distortion is a warping or transformation of an object and its surrounding area that differs significantly from what the object would look like with a normal focal length, due to the relative scale of nearby and distant features. Perspective distortion is determined by the relative distances at which the image is captured and viewed, and is due to the angle of view of the image being either wider or narrower than the angle of view at which the image is viewed, hence the apparent relative distances differing from what is expected. Related to this concept is axial magnification -- the perceived depth of objects at a given magnification.

In optics, the Airy disk and Airy pattern are descriptions of the best-focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light. The Airy disk is of importance in physics, optics, and astronomy.

Magnification is the process of enlarging the apparent size, not physical size, of something. This enlargement is quantified by a calculated number also called "magnification". When this number is less than one, it refers to a reduction in size, sometimes called minification or de-magnification.

An eyepiece, or ocular lens, is a type of lens that is attached to a variety of optical devices such as telescopes and microscopes. It is so named because it is usually the lens that is closest to the eye when someone looks through the device. The objective lens or mirror collects light and brings it to focus creating an image. The eyepiece is placed near the focal point of the objective to magnify this image. The amount of magnification depends on the focal length of the eyepiece.

The Scheimpflug principle is a description of the geometric relationship between the orientation of the plane of focus, the lens plane, and the image plane of an optical system when the lens plane is not parallel to the image plane. It is applicable to the use of some camera movements on a view camera. It is also the principle used in corneal pachymetry, the mapping of corneal topography, done prior to refractive eye surgery such as LASIK, and used for early detection of keratoconus. The principle is named after Austrian army Captain Theodor Scheimpflug, who used it in devising a systematic method and apparatus for correcting perspective distortion in aerial photographs; even though Captain Scheimpflug himself credits Jules Carpentier with the rule, thus making it an example of Stigler's law of eponymy.

The Cassegrain reflector is a combination of a primary concave mirror and a secondary convex mirror, often used in optical telescopes and radio antennas, the main characteristic being that the optical path folds back onto itself, relative to the optical system's primary mirror entrance aperture. This design puts the focal point at a convenient location behind the primary mirror and the convex secondary adds a telephoto effect creating a much longer focal length in a mechanically short system.

Depth of focus is a lens optics concept that measures the tolerance of placement of the image plane in relation to the lens. In a camera, depth of focus indicates the tolerance of the film's displacement within the camera and is therefore sometimes referred to as "lens-to-film tolerance".

In Gaussian optics, the cardinal points consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the focal points, the principal points, and the nodal points. For ideal systems, the basic imaging properties such as image size, location, and orientation are completely determined by the locations of the cardinal points; in fact only four points are necessary: the focal points and either the principal or nodal points. The only ideal system that has been achieved in practice is the plane mirror, however the cardinal points are widely used to approximate the behavior of real optical systems. Cardinal points provide a way to analytically simplify a system with many components, allowing the imaging characteristics of the system to be approximately determined with simple calculations.

A curved mirror is a mirror with a curved reflecting surface. The surface may be either convex or concave. Most curved mirrors have surfaces that are shaped like part of a sphere, but other shapes are sometimes used in optical devices. The most common non-spherical type are parabolic reflectors, found in optical devices such as reflecting telescopes that need to image distant objects, since spherical mirror systems, like spherical lenses, suffer from spherical aberration. Distorting mirrors are used for entertainment. They have convex and concave regions that produce deliberately distorted images. They also provide highly magnified or highly diminished (smaller) images when the object is placed at certain distances.

Vergence is the angle formed by rays of light that are not perfectly parallel to one another. Rays that move closer to the optical axis as they propagate are said to be converging, while rays that move away from the axis are diverging. These imaginary rays are always perpendicular to the wavefront of the light, thus the vergence of the light is directly related to the radii of curvature of the wavefronts. A convex lens or concave mirror will cause parallel rays to focus, converging toward a point. Beyond that focal point, the rays diverge. Conversely, a concave lens or convex mirror will cause parallel rays to diverge.

Petzval field curvature, named for Joseph Petzval, describes the optical aberration in which a flat object normal to the optical axis cannot be brought properly into focus on a flat image plane.

## References

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